Question: Is ${207774}$ divisible by $9$ ?
Explanation: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {207774}= &&{2}\cdot100000+ \\&&{0}\cdot10000+ \\&&{7}\cdot1000+ \\&&{7}\cdot100+ \\&&{7}\cdot10+ \\&&{4}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {207774}= &&{2}(99999+1)+ \\&&{0}(9999+1)+ \\&&{7}(999+1)+ \\&&{7}(99+1)+ \\&&{7}(9+1)+ \\&&{4} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {207774}= &&\gray{2\cdot99999}+ \\&&\gray{0\cdot9999}+ \\&&\gray{7\cdot999}+ \\&&\gray{7\cdot99}+ \\&&\gray{7\cdot9}+ \\&& {2}+{0}+{7}+{7}+{7}+{4} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${207774}$ is divisible by $9$ if ${ 2}+{0}+{7}+{7}+{7}+{4}$ is divisible by $9$ Add the digits of ${207774}$ $ {2}+{0}+{7}+{7}+{7}+{4} = {27} $ If ${27}$ is divisible by $9$ , then ${207774}$ must also be divisible by $9$ ${27}$ is divisible by $9$, therefore ${207774}$ must also be divisible by $9$.